A graph G has the pigeonhole property, written (P), if
whenever the vertex set is partitioned into two parts, then the
subgraph induced by one of the parts is isomorphic to G. In 1996,
Peter Cameron proved the surprising result that the only countable
graphs with (P) are the trivial graph, the countably
infinite complete graph and its complement, and the countably infinite
random graph. Last summer, at a conference in honour of Roland
Fraïssé, Cameron posed the following problem: which countable
graphs G have the property that whenever the vertex set is
partitioned into three parts, the subgraph induced by the union of some
two of the parts is isomorphic to G? This new vertex partition
property is called P(3,2).

In this talk, we answer Cameron's problem and present a classification
of the countable graphs with P(3,2). A classification of
the countable digraphs with P(3,2) will also be presented.
This is joint work with Peter Cameron, Dejan Deli\'c, and Stéphan
Thomassé.

In 1973, J. Nesetril and V. Rödl asked whether, for k³ 4, a large k-critical graph necessarily contains a large
(k-1)-subgraph. The statement is true for k=4 (see references in
Problem 5.6 of B. Toft & T. Jensen's book Graph Coloring
Problems). In this talk, I will describe a construction for k=5,6,
that shows there exists arbitrarily large k-critical graphs on n
vertices all of whose (k-1)-critical subgraphs have O(logn)
vertices.

Convex-round (concave-round) graphs are those graphs whose vertices can
be circularly enumerated so that the (closed) neighbourhood of each
vertex forms an interval in the enumeration. These two classes
transform into each other by taking complementary graphs. The class of
concave-round graphs properly contains the class of proper circular arc
graphs and is properly contained in the class of general circular arc
graphs. We show that both convex-round and concave-round graphs have
nice structural properties and that they can be recognized in O(n+m)
time (here n denotes the number of vertices and m the number of
edges of the graph in question). We show that the chromatic number of
a graph which is convex-round (concave-round) can be found in time
O(n+m) (O(n2)). We describe optimal O(n+m) time algorithms for
finding a maximum clique, a maximum matching, and a hamiltonian cycle
(if one exists), for the class of convex-round graphs. Finally, we
show that all convex-round graphs are circular perfect.

We will discuss several results on embedding graphs in Euclidean
spaces. Applications include constructions of large few-distance sets
in Euclidean spaces and graph-theoretic aspects of the Borsuk
conjecture.

In this talk, we investigate the total colorings of the join graphs
G1 + G2, where G1 and G2 are graphs with maximum degree at
most two. We prove that (1) when both G1 and G2 are bipartite
graphs with maximum degree at most two, then G is Type 1 if and
only if G is not isomorphic to Kn,n (n = 1,2,¼) or to
K4, and (2) Cm + Cn is Type 2 if and only if m = n and n is
odd.

A fundamental problem in the theory of error-correcting digital codes
is to determine the asymptotic tradeoff between robustness and message
expansion in block codes. The Varshamov-Gilbert bound guarantees, via
the probabilistic method, that good codes exist as the length goes to
infinity while the linear programming bound of Delsarte is the best
known technique for showing that codes do not exist. A recent result of
A. Samorodnitsky shows that these two quantities diverge. Thus new
constructions are called for and new general techniques for
establishing non-existence are needed.

In this talk, we restrict attention to the binary case. The Delsarte
bound is based on the structure of the adjacency algebra of the
n-cube. The Terwilliger algebra is a non-commutative
extension of the adjacency algebra which contains more refined
information. We propose this algebra as a mode of investigation for
improved upper bounds on the efficiency of binary codes.

Let Aj denote the adjacency matrix of the distance-i graph of the
n-cube and let Ei* denote the diagonal matrix having a one in
position (u,u) if u is a binary tuple of Hamming weight i and
zeros elsewhere. The matrices Ei*AjEk* (omitting copies of
the all-zero matrix) give a basis for the Terwilliger algebra. For a
code C with characteristic vector c, the statistics cTEi*AjEk*c form the coefficients of the biweight
enumerator of C which was introduced by MacWilliams, et al.
in 1972 and has been studied in relation to linear codes. We show how
the Terwilliger algebra sheds new light on this enumerator and how this
can be used to obtain some improved bounds on the efficiency of binary
codes. The tools we use include graph theory and linear algebra.

A circulant graph is a Cayley graph X(n;S) on the cyclic group of
order n, where S is an inverse-closed subset of Zn that
does not contain 0. This graph consists of n vertices labeled 0,¼, n-1, where the vertex labeled i is connected by an edge to
the vertex labeled j precisely if i-j mod n is in S. The
automorphism group of this graph, consisting of all graph isomorphisms
from the graph to itself, is denoted by Aut(X).

It is clear that the mapping taking i®i+j for all i and
for any fixed j is an automorphism of this graph. Call this
automorphism aj. A normal circulant graph is one for which the
cyclic automorphism subgroup consisting of {aj:0 £j£n-1} is normal in the full automorphism group. The normal circulant
graphs that can also be represented as Cayley graphs on some non-cyclic
group of order n are characterised. This result is compared with
other results on graphs that can be represented as Cayley graphs on
distinct groups of order n. This is based on joint work with
Dr. Dragan Marusic of the University of Ljubljana.

A weighting f: V(G)®R of a graph G is
well-covered if the sum of the weights is the same for each maximal
independent set of G. A graph is well-covered if the all-ones vector
is a well-covered weighting. Caro et al. show that well-covered
weightings form a vector space. We follow up the problems of:
determining the dimension of this space for families of graphs;
determining the non-zero weighted vertices in a basis vector; and
indicate what this implies for the problem of characterizing
well-covered graphs in general.

In this joint work with Carsten Thomassen, we prove that if K is a
3-connected, locally connected subset of the sphere S2 and if
f: K®S2 is any embedding of K in S2, then there is a
homeomorphism h: S2®S2 such that h restricted to K is
f. This generalizes the well-known fact that 3-connected graphs
have unique embeddings in the sphere. A particular application is that
if G is a connected, locally finite graph, then the Freudenthal
compactification of G (which adds one point to G for every end of
G) has a unique embedding in the sphere.

A cycle double cover of a graph G is a collection of cycles,
C, such that every edge of G lies in precisely two cycles
of C. The Small Cycle Double Cover (SCDC)
Conjecture, proposed by J. A. Bondy, asserts that every simple
bridgeless graph on n vertices has a cycle double cover with at most
n-1 cycles, and is a strengthening of the well-known Cycle
Double Cover (CDC) Conjecture.

Both the CDC Conjecture and the SCDC Conjecture have been verified for
various classes of graphs, but remain open in general. The graphs for
which that SCDC Conjecture has been verified (including simple
triangulations of surfaces, 4-connected planar graphs, and line
graphs of numerous types of graphs) all have well defined structural
properties that play an important role. The structure that is inherent
in graph products makes such graphs ideal special cases for which to
verify the SCDC Conjecture. There are various graph products that can
be considered, and in this talk I will describe some results and
techniques for proving the SCDC Conjecture for certain graph products.
This is joint work with R. J. Nowakowski.

Hedetniemi's conjecture states that the chromatic number of a product
of two graphs is equal to the minimum of the chromatic numbers of the
factors. The inherent difficulty lies in finding lower bounds for the
chromatic number of a product of two graphs. In fact, it is not yet
known if there exists a number M such that the chromatic number of
the product of any two M-chromatic graphs is at least 5. In this
talk, we show that the chromatic number of a product of two graphs is
at least half the minimum of the fractional chromatic numbers of the
factors.

Let G be a graph. If each k-matching of G can be extended to a
perfect matching, then G is called k-extendable (edge
version of extension). If after deleting any n vertices the remaining
subgraph of G has a perfect matching, then G is called n-factor-critical (vertex version of extension). We will survey
the recent progress on the matching extension problem and also discuss
the other extensions and the relationship among them.